Notice that the artificial inertial parameter \(m_0\)  in the Lagrangian dynamics (REF )-(REF ) is related with the multiplier in the Euler-Lagrange modelling [1]}, [2]}. Since the choice of \(m_0\)  or the Lagrangian multiplier is not unique, the acceleration of the quaternion should not depend on these parameters. The independence of the Lagrangian multiplier in the Lagrangian dynamics was shown in [2]}. The Lagrangian dynamics (REF ) is also independent of \(m_0\) . In fact, by the Lagrangian dynamics (REF ), the fact of \(D^{-1}(q)=Q(q)M_0^{-1}Q^T (q)\)  and \(q^T\dot{q}=0\) , and some manipulations it follows that
\(\ddot{q} &=D^{-1}(q)\Big ( -C(q,\dot{q})\dot{q}+\bar{\tau }\Big )  \\&=J(q)M^{-1}S(M\omega )J^{T}(q)\dot{q} +J(\dot{q})J^T(q)\dot{q} +\frac{1}{2}J(q)M^{-1}\tau \) 
